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Phys410, Classical Mechanics Notes Ted Jacobson Phys410, Classical Mechanics Notes University of Maryland, College Park Ted Jacobson December 20, 2012 These notes are an evolving document... Contents 1 Energy 3 1.1 Potential energy . 3 2 Variational calculus 4 2.1 Euler-Lagrange equations . 4 3 Lagrangian Mechanics 4 3.1 Constraints . 6 3.2 Effective potential; example of spherical pendulum . 8 3.3 Spinning hoop . 9 4 Hamiltonian and Conservation of energy 9 5 Properties of the action 10 6 Electromagnetic field 12 6.1 Scalar and vector potentials . 12 6.2 Lagrangian for charge in an electromagnetic field . 13 1 7 Lagrange multipliers and constraints 14 8 Tidal force and potential 14 9 Velocity in a rotating frame 15 10 Special Relativity 16 10.1 Spacetime interval . 18 10.1.1 Proper time . 18 10.1.2 Pythagorean theorem of spacetime . 18 10.1.3 Time dilation and the twin effect . 20 10.1.4 Spacetime geometry . 20 10.1.5 Velocity and time dilation . 22 10.2 Inertial motion . 22 10.3 Action . 24 10.4 Energy and momentum . 25 10.4.1 Relation between energy and momentum . 26 10.4.2 Massless particles . 26 10.5 4-vectors and the Minkowski scalar product . 27 10.5.1 Conventions . 28 10.5.2 \Look Ma, no Lorentz transformations" . 28 10.6 4-momentum and 4-velocity . 28 10.6.1 Pesky factors of c ..................... 30 10.6.2 Example: Relativistic Doppler effect . 30 10.7 Zero momentum frame . 31 10.7.1 Example: Head-on vs. fixed target collision energy . 31 10.7.2 Example: GZK cosmic ray cutoff . 32 10.8 Electromagnetic coupling . 33 11 General relativity 34 12 Hamiltonian formalism 37 12.1 Hamilton's equations and the Hamiltonian . 38 12.2 Example: Bead on a rotating circular hoop . 39 12.2.1 Driven hoop . 40 12.2.2 Freely rotating hoop . 41 12.3 Example: Charged particle in an electromagnetic field . 42 12.4 Phase space volume and Liouville's theorem . 43 2 12.5 Phase space and quantum mechanics . 44 12.6 Entropy and phase space . 45 12.7 Adiabatic invariants . 46 12.7.1 Example: Oscillator with slowly varying m(t) and k(t) 48 12.7.2 Adiabatic invariance and quantum mechanics . 49 12.7.3 Example: Cyclotron orbits and magnetic mirror . 49 12.8 Poisson brackets . 49 12.8.1 Canonical Quantization . 51 12.8.2 Canonical transformations . 51 13 Continuum mechanics 52 13.1 String . 53 13.2 Electromagnetic field . 55 13.3 Elastic solids . 55 1 Energy 1.1 Potential energy The concept of potential energy arises by considering forces that are (minus) the gradient of a function, the "potential". For such forces, if the potential is time independent, the force is said to be "conservative", and the work along a path is just minus the change of the potential, thanks to the fundamental theorem of calculus applied to line integrals. The work for such a force is therefore independent of the path that connects two given endpoints. By Stokes' theorem, this is related to the fact that the curl of such a force is zero, since the curl of the gradient of anything is zero. Central forces F = f(r)^r are derivable from a potential. The key is that rr = ^r, which I explained both computationally and in terms of the geomet- rical interpretation of the gradient: it points in the direciton of greatest rate of change of the function, and has magnitude equal to that rate of change. Thus we can write Z r f(r)^r = f(r)rr = r dr0f(r0) (1) which shows that the potential for this radial force is U(r) = U(r) = − R r f(r0)dr0. 3 2 Variational calculus 2.1 Euler-Lagrange equations I explained the nature of a \functional" and what it means for that to be stationary with respect to variations of the function(s) that form its argu- ment. As an alternative to the method described in the book, I re-derived the Euler-Lagrange equations without introducing any particular path variation eta. - Examples: soap film stretched between hoops, length of a curve in the Euclidean plane. We solved this three ways: 1) paths y(x) [could instead take x(y)] 2) parametrized paths x(t), y(t) 3) parametrized paths r(t), θ(t) using the E-L equations. In the second case, we noted that the path param- eter has not been specified, so there is no reason whyx _(t) andy _(t) should be constant. But we found thatx _(t)=y_(t) is constant, which implies that dx=dy (or dy=dx) is constant. In the 3rd case, the eqns are complicated, but if we use the translation symmetry to place the origin of the coordinate system on the curve, we see that the theta equation implies θ_ = 0, which is certainly the description of a straight line through the origin. 3 Lagrangian Mechanics Pulled out of a hat the definition of the Lagrangian, L = T − U, and the action S = R L dt, also called Hamilton's principal function. Showed that for a particle in 1d the condition that S be stationary under all path variations that vanish at the endpoints is equivalent to Newton's second law. This is called \Hamilton's principle". Then generalized this to a particle in 3d, then to two particles in 3d interacting with each other via a potential. It general- izes to any number of particles. - It's quite remarkable that the collection of vector equations of a system of a system of particles all come from Hamilton's principle, which refers to the variation of the integral of a scalar. Adding more particles or dimensions 4 increases the number of functions that the action depends on, but it's still the integral of a scalar. - Although it looks arbitrary at first, the action approach is actually the deeper approach to mechanics. The action approach also governs relativistic mechanics, and even field theory. For example Maxwell's equations and even Einstein's field equations of gravitation are all governed by an action princi- ple. In the case of fields, the Lagrangian is an integral over space. Moreover, it is via the action that the role of symmetries is best understood and ex- ploited. Also, as a practical matter, one of the most powerful things about the Lagrangian formalism is the flexibility of the choice of variables, since by using variables adapted to a system one can simplify the equations. Choice of variables can also be useful in exploiting approximation schemes. - The significance of the action and Hamilton's principle can be under- stood from the viewpoint of quantum mechanics. In Feynman's path integral formulation, each path is assigned the amplitude exp(iS=h¯), where h¯ is Planck's constant. (It only makes sense to exponentiate a dimensionless quantity. S has dimensions of action = (energy) × (time) = (momentum) × (length), the same ash ¯.) The total amplitude is the sum over all paths. Destructive interference occurs when the action of two paths differs by some- thing comparable toh ¯ or greater. This is howh ¯ sets the scale of quantum effects. At the classical path, the variation of S vanishes, so nearby paths interfere constructively. In the classical limit, the path is thus determined by the condition that S be stationary. You can read about this in the Feynman lectures, for instance. R 1 2 - What is action? For a free particle motion the action is S = 2 mv dt, which is the average kinetic energy times the total time interval. On the classical path (solution to the equation of motion) v = v0 = const. We can easily show this is the minimum for all paths. In the presence of a potential, the action is still a minimum on the classical path, provided the two times are close enough. For a harmonic oscillator, "short enough" means less than half the period. - Can change variables freely in describing the configuration of the sys- tem. Example: change from (x1; x2) to (xcm; xrel). 5 3.1 Constraints When the configuration coordinates of a system are constrained by physical conditions, then one can just impose the constraint in the the Lagrangian, eliminating a constrained degree of freedom and omitting the potential that enforces the constraint. This is correct because after imposing the constraint, although the variations of the original coordinates are restricted, they are all valid variations, so the action must be stationary with respect to them, so the corresponding E-L equations must hold. If there are enough E-L equations to determine the time evolution of the remaining coordinates, then the description is complete. Let's illustrate this with the example of a simple pendulum hanging from a string of fixed length. In terms of spherical coordinates based at the vertex, the mass can move freely in θ and φ, but the r degree of freedom is constrained to be equal to a fixed length r0 by some constraining potential Uconst(r) arising from the microscopic structure of the string. If r is set equal to r0 in the Lagrangian, the θ and φ equations remain valid and they determine the evolution of these coordinates. A more explicit argument goes as follows. The full Lagrangian is 1 2 2 _2 2 2 _2 L = 2 m(_r + r θ + r sin θ φ ) + mgr cos θ − Uconst(r): (2) The Lagrange equation for r is 0 mr¨ = mg cos θ − Uconst(r): (3) If we know that the constraint is satisfied at r = r0, then we can just omit Uconst(r) and set r equal to r0 in the Lagrangian.
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